A new lifetime model by mixing gamma and geometric distributions useful in hydrology data

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1 ProbStat Forum, Volume 2, January 29, Pages 4 ISSN ProbStat Forum is an e-journal. For details please visit A new lifetime model by mixing gamma and geometric distributions useful in hydrology data Hassan S. Bakouch a, Abdus Saboor b, Christophe Chesneau c, Anwaar Saeed b a Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt. b Department of Mathematics, Kohat University of Science & Technology, Kohat, Pakistan. c LMNO, University of Caen, France. Abstract. In this paper, we propose a new distribution obtained by mixing gamma and geometric distributions. We discuss different shapes of the probability density function and the hazard rate functions. We study several statistical properties. The maximum likelihood estimation method is performed for estimating the parameters. We determine the observed information matrix and discuss inference. Illustrative three hydrology data sets are given to show the flexibility and potentiality of the proposed distribution.. Introduction An important aspect of statistics is the determination of flexible distributions to elaborate useful models for lifetime data. Among the existing approaches, new distributions can be obtained by mixing discrete and continuous distributions. Those using geometric distributions include the exponential geometric distribution [], the exponential-power series distribution [9], the extended exponential geometric distribution [2], the complementary exponential geometric distribution [4], the Weibull-geometric distribution [6], the Weibullpower series distribution [8], the generalized exponential-power series distribution [6], the complementary exponentiated exponential geometric distribution [5], the extended Weibull-power series distribution [2], the complementary extended Weibull-power series distribution [], the exponentiated extended Weibullpower series distribution [23], the G-geometric distribution [3], the alternative G-geometric distribution [8], the generalized linear failure rate-geometric distribution [2] and the linear failure rate-power series distribution [7]. We also refer to the review of [22], and the references therein. On the other side, among the continuous distributions, the gamma distribution is one of the most commonly used in modeling life-time data. In practice, it has been shown to be very flexible in modeling various types of lifetime distributions. To the best of our knowledge, the mixing of the geometric distribution with the gamma distribution not reduced to the exponential one has not ever been considered in the literature. Based on such a mixing, this paper offers a new distribution with two parameters, called the gamma-geometric GG distribution. The formulation and motivations of such distribution are as follows. Let λ >, θ, and θ = θ. We say that a random variable X follows the GG distribution with 2 Mathematics Subject Classification. MSC 2: 2E, 62E5, 62F. Keywords. Gamma distribution, Geometric distribution, Order statistics, Record values. Received: 7 May 28; Revised: 25 January 29, Accepted: 3 January 29. addresses: hnbakouch@yahoo.com Hassan S. Bakouch, saboorhangu@gmail.com Abdus Saboor, christophe.chesneau@unicaen.fr3 Christophe Chesneau, Anwaarsaeedqureshi@gmail.com Anwaar Saeed

2 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 2 parameters θ, λ, denoted by GGθ, λ, if it has the probability density function pdf given by fx = θλ 2 xe + θe θe 3, x >. The corresponding cumulative distribution function cdf is given by + λx θe F x = θe θe, x >. 2 2 The GG distribution arises from the following stochastic representation. Let X be a random variable having the following stochastic representation: X {N = n} G am 2, λn, N θ G trunc θ, 3 that is N is a random variable having the truncated geometric G trunc distribution with parameter θ: P N = n = θ θ n, n =, 2,... and the distribution of X conditionally to {N = n} is the gamma distribution G am 2, λn, with a conditional pdf given by f X {N=n} x = λ 2 n 2 xe λnx, x >. Then X follows the GGθ, λ distribution; using the geometric series expansion: + n 2 x n = x+x x, x <, the pdf of X is given by 3 f X x = f X {N=n} xp N = n = θ θ λ 2 x n 2 θe n = θλ 2 xe + θe θe 3. 4 The stochastic representation 3 can be viewed as a natural extension of the stochastic representation X {N = n} E xp λn = G am, λn, with pdf corresponding to the one of the G-geometric class proposed by [3] applied with the exponential distribution. Ratio of two independent variables. An example of simple model using the GG distribution is given by the ratio of two independent variables as described as follows. Let Y G am 2, λ and N G trunc θ. Suppose that Y and N are independent. Then the ratio of Y and N given by X = Y N, follows the GGθ, λ distribution. It is enough to note that X {N = n} = Y/n G am 2, λn. This ratio representation will be useful to determine statistical properties of the GG distribution. Note that, from the ratio representation, the cdf of X given by 2 can be expressed directly: using the cdf of Y : F Y x = e λxe, x >, and the geometric series expansions: + x n = x and nx n = x 2, x <, the cdf of X is given by F X x = P Y xn = F Y xnp N = n = θe θe n λxθe n= + λx θe = θe θe. 2 Some limit properties for fx are given as: n θe n fx λ 2 2 θ θ 2 x, x, fx θλ 2 xe, x +. n=

3 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages Λ=.5 Θ=.75 Θ=.9 Θ=.2 Θ=.3 Θ= Λ= Θ=.75 Θ=.9 Θ=.2 Θ=.3 Θ= Figure : Plots of the GG density function. Moreover, one can show that fx has a unique maximum on, + given by fx where x satisfies the equation: 4 θλx e λx + λx e 2λx + θ 2 λx + =. Some plots of fx are given in Figure for several values of θ, λ. The rest of the paper is organized as follows. In Section 2, we give some properties of the GG distribution. The estimation by maximum likelihood is discussed in Section 3. Three illustrative real-life data examples are provided in Section Properties of the GG distribution In this section, we propose many features and statistical properties of the GG distribution. 2.. The survival and hazard rate functions The survival function sf of X is given by + λx θe Sx = F x = θe θe, x >, 5 2 and the associated hazard rate function hrf of X is θe hx = fx Sx = + λ2 x θe + λx θe, x >. 6 Observe that hx λ 2 2 θ θ 2 x, x, hx λ, x +. Some plots of hx are given in Figure 2 for several values of θ, λ Quantile function The quantile function of X is determined by inverting the cdf F x. The p-th quantile x p of X is the solution of the nonlinear equation: F x p = p θe p + λx p θe p = p θe p 2.

4 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages Λ=.5 Θ=.75 Θ=.9 Θ=. Θ=.2 Θ= Λ= Θ=.75 Θ=.9 Θ=.2 Θ=.3 Θ= Figure 2: Plots of the GG hazard rate function Moments Some key features of a distribution, like mean and variance, can be investigated through its r-th moments EX r. For finding EX r, we can use the ratio representation of X: X has the same distribution of the ratio of 2 random variables: Y/N with Y G am 2, λ and N G trunc θ, Y and N independent. Therefore, considering the Gamma function: Γν = + x ν e x dx, ν >, E Y r = Γ2+r λ and the polylogarithm r function: Li r x = + x n, r >, x <, we have nr Y EX r r = E N r = E Y r Γ2 + r E N r = λ r In particular, by taking r =, since Li x = log x, we obtain The variance of X can be explicit in some cases. Li 2 2 = 2 [π2 6log2 2 ], we have V arx = EX 2 [EX] 2 = Γ4 λ 2 Li 2 n EX = 2θ λ θ logθ. λ r r P N = n = Γ2 + r θ θ Li r θ. For instance, if θ = 2, since Li 2 = log2 and [ Γ3 2 λ Li ] 2 = π 2 2 λ 2 2 7log Moment generating function Also the moment generating function of X can be obtain via the ratio representation Y/N. Ee ty λ 2 =, t < λ, and the conditional expectation, we get λ t 2 Mt = Ee tx = Ee t Y N = E E e t Y N 2 N N = E = λ 2 + n 2 λn t 2 P N = n = n 2 λ2 θ θ n λn t 2. λ 2 λ t 2 = λ 2 E N λn t 2 Using

5 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages Conditional and reversed moments The r-th conditional moments of X is given by EX r X > t = EX r St and the r-th reversed moments of X is given by EX r X t = F t x r fxdx, t >. x r fxdx, t >, The integral term can be expressed using the expansion 4. gamma function Γt, ν = xν e x dx, ν >, t >, we have Indeed, introducing the lower incomplete x r fxdx = θ θ + λ 2 n 2 θn x r+ e nλx dx = θ θ λ r θ n Γnλt, r nr 2.6. Rényi entropy An entropy plays a central role in information theory. It provides a suitable measure of randomness or uncertainty of X. For continuous distributions, Rényi entropy see [2] can be determined as follows: We have I R γ = + γ log [fx] γ dx, γ >, γ. [fx] γ = θ γ λ 2γ x γ e λγx + θe γ θe 3γ. Using the generalized binomial series: + x α = + have Hence, + θe γ = [fx] γ = θ γ λ 2γ + l= k= k= 3γ γ θ k e λkx, k l k= α k x k, α C, x <, α k = αα α 2...α k+ k!, we + θe 3γ = γ l k θl+k x γ e λl+k+γx. l= 3γ l l θl e λlx. Therefore, I R γ = = [ + γ logθ + 2γ logλ + log γ l= [ + γ logθ + 2γ logλ + log γ k= l= k= 3γ γ l k 3γ γ l k l θl+k + x γ e λl+k+γx dx l θl+k Γγ + λ γ+ l + k + γ γ+ ] ].

6 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages Order statistics distributions The order statistics are central tools in non-parametric statistics and inference. Let us now present the distributions of some fundamental order statistics related to the GGθ, λ distribution. Let a sample X, X 2,..., X n is randomly chosen from the GGθ, λ distribution and X :n X 2:n... X n:n are its corresponding order statistics. A pdf of X i:n is given by f Xi:n x = = n i n! i! n i! fx l= n i n! i! n i! θλ2 + θe xe θe 3 x >. The cdf of X i:n is given by l l [F x] i +l n i n i [ l θe l l= + λx θe θe 2 ] i +l, F Xi:n x = = n! i! n i! n! i! n i! n i l n i [F x]i+l l i + l l= n i l n i [ θe l i + l l= + λx θe θe 2 ] i+l, x >. A joint pdf of X :n,..., X n:n is given by f X:n,...,X n:nx,..., x n = n! n n fx k = n!θ n λ 2n x k k= A joint pdf of X i:n, X j:n, i < j, is given by < x <... < x n. k= e λ x k k= n + θe k k= n, θe 3 k n! f Xi:n,X j:nx i, x j = i! n j! j i [F x i] i [F x j F x i ] j i [Sx j ] n j fx i fx j [ n! = θe + λx ] i i i i θe i! n j! j i θe i 2 [ θe + λx i i i θe θe θe + λx ] j j i j j θe i 2 θe j 2 [ θe + λx ] j n j j j θe θe θ 2 λ 4 x j 2 i x j e λxi+xj i j + θe + θe θe 3 i θe 3, j < x i < x j. k= 2.8. Record values distributions Record values arise in a wide varity of real-life applications as hydrology, industry, lifetesting, economics, among the others. See, for instance, [4], [5] and []. We now present important distributions related to record values using the GGθ, λ distribution as baseline. Let X, X 2,..., be a sequence of i.i.d. random variables having the GGθ, λ distribution. We define a sequence of record times Un as follows: U =, Un = min{j; j > Un, X j > X Un } for n 2. We define the i-th upper record value by R i = X Ui,

7 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 7 with R = X. A pdf of R i is given by f Ri x = i! [ logsx]i fx [ = log θe i! A joint pdf of R,..., R n is given by + λx θe θe 2 n f R,...,R nx,..., x n = fx n hx k = θλ 2n x n e n n k= k= x k A joint pdf of R i, R j, i < j, is given by n ] i θλ 2 + θe xe θe 3, x >. + θe k n k= θe n + θe 3 n, < x <... < x n. θe k + λxk θe k k= f Ri,R jx i, x j = i!j i! [ logsx i] i [logsx i logsx j ] j i hx i fx j [ = log θe + λx ] i i i i θe i!j i! θe i 2 [ log θe + λx i i i θe θe log θe + λx ] j j i j j θe i 2 θe j 2 θe i θe j λ 2 + x i θe i + λx i θe i + θλ2 x j e j θe 3, < x i < x j. j 2.9. Residuals life functions The residual life functions play a fundamental role in survival or reliability studies. See, for instance, [7], [3] and [9]. We now present some related mathematical objects with a potential of interest in the context of the GGθ, λ distribution. The residual life is described by the conditional random variable R t = X t {X > t}, t. The sf of the residual lifetime R t is given by S Rt x = Sx + t St The associated cdf is given by = e + λx + t θe λx+t θe λt 2 θe λx+t 2 + λt θe, x >. λt F Rt x = e + λx + t θe λx+t θe λt 2 θe λx+t 2 + λt θe, x >. λt Then, the corresponding pdf is given by f Rt x = λ 2 x + te + θe λx+t θe λt 2 θe λx+t 3 + λt θe λt, x >. The associated hrf is given by θe λx+t h Rt x = λ 2 + x + t θe λx+t + λx + t θe λx+t x >,

8 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 8 and the mean residual life is defined as Kt = ER t = EX t {X > t} = St EX The integral term can be expressed as 7 with r =. On the other side, the variance residual life is given by V t = V arr t = V arx t {X > t} = St EX 2 xfxdx t. x 2 fxdx t 2 2tKt [Kt] 2. Again, the integral term can be expressed as 7 with r = 2. The reverse residual life is described by the conditional random variable R t = t X {X t}, t. The sf of the reversed residual lifetime R t is given by S Rt x = F t x F t The associated cdf is given by = θe λt 2 [ θe λt x 2 θe λt x + λt x θe λt x ] θe λt x 2 [ θe λt 2 θe λt + λt θe λt ], < x t. F Rt x = θe λt 2 [ θe λt x 2 θe λt x + λt x θe λt x ] θe λt x 2 [ θe λt 2 θe λt + λt θe λt ], < x t. Therefore, the corresponding pdf is given by f Rt x = θλ 2 t xe λt x + θe λt x θe λt 2 θe λt x 3 [ θe λt 2 θe λt + λt θe λt ], < x t, and the associated hrf is given by h Rt x = θλ 2 t xe λt x + θe λt x θe λt x 2 θe λt x 3 [ θe λt x 2 θe λt x + λt x θe λt x ], < x t. The mean reversed residual life is defined as Lt = ER t = Et X {X t} = t F t The integral term can be expressed as 7 with r =. The variance reversed residual life is given by xfxdx. W t = V arr t = V art X {X t} = 2tLt [Lt] 2 t 2 + F t Again, the last integral can be expressed as 7 with r = 2. x 2 fxdx. 3. Maximum likelihood estimation In this section, we estimate the unknown parameters of the GG distribution using the method of maximum likelihood. Let X, X 2,..., X n be a random sample of size n from the GGθ, λ distribution with observed values x, x 2,..., x n. Set Θ = {θ, λ}. The likelihood function associated to x,..., x n is given by LΘ = n fx i = i= n θλ 2 i n + θe x i e i θe 3 = θ n λ 2n x i i i= i= e λ x i i= i= n e λx i + θ n. e λxi + θ 3 i=

9 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 9 The maximum likelihood estimators MLEs of θ and λ are obtained by maximization of LΘ, or alternatively, the log-likelihhod defined by lθ = loglθ = n logθ + 2n logλ + + i= logx i + λ i= loge λxi + θ 3 loge λxi + θ. i= It follows that the MLEs are the simultaneous solutions of the equations according to θ, λ: and lθ θ lθ λ = n θ n i= = 2n n λ + x i + i= n e λxi + θ 3 e λxi + θ = i= i= i= e λxi x n i e λxi + θ 3 e λxi x i e λxi + θ =. i= Since we have no analytic forms, numerical methods, as the quasi-newton algorithm, can be applied to determine the estimators. The observed information matrix is given by Jθ JΘ = θ Θ J θ λ Θ, J λ θ Θ J λ λ Θ where J θ θ Θ = 2 lθ θ 2 J θ λ Θ = 2 lθ θ λ J λ λ Θ = 2 lθ λ 2 3 i= = n θ 2 + n i= i= n e λxi + θ 2 3 e λxi + θ 2, n = e λxi x n i e λxi + θ 2 3 = 2n λ 2 + n i= i= i= x i e λxi x i e λxi + θ 2, e 2λxi x 2 i e λxi + θ 2 eλxi x 2 i e 2λxi x 2 i e λxi + θ 2 e λxi x 2 i e λxi + θ. e λxi + θ This matrix is a key mathematical tool to obtain approximate confidence intervals or Wald tests for θ and λ in the case of a large sample. 4. Illustrative hydrology data examples In this section, we take three hydrology data sets to show the flexibility and potentiality of the proposed distribution. We fit the GG distribution to three hydrologic data sets and compare with the Weibull, Gumbel, Exponentiated Exponential, Generalized Gumbel, Kappa and Weibull Geometric distributions for three data sets. Most of those distributions have received great attention for fitting hydrology data, like rainfall data, precipitation data and flood data. More precisely, the densities of the compared distributions are given as follows: Weibull distribution with pdf: fx = k x k e λ x k, λ >, k >, x >. λ λ

10 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 Gumbel distribution with pdf: fx = e e x µ σ x µ σ σ, σ >, x, µ R. Exponentiated Exponential EE distribution [24] with pdf: fx = αλ e α e, α, λ, x >. Generalized Gumbel GGu distribution [25] with pdf: fx = x µ e σ α e Kappa distribution [26] with pdf: fx = αθ β θ x α + β α e x µ x µ σ e σ e σ, α, σ >, µ, x R. α+ αθ α x, α, θ, β, x >. β Weibull geometric WG distribution [6] with pdf: fx = αβ α px α e βxα p e βxα 2, p,, α, β, x >. For goodness-of-fit we have two main test statistics, i.e., information criterion and empirical distribution. The measures Akaike information criterion AIC [27], corrected Akaike information criterion AICC [28], Hannan Quinn information criterion HQIC [29], and consistent Akaike information criterion CAIC [3] are widely used information criterion for selecting the appropriate model among other models. The Anderson-Darling A due to Anderson and Darling [3], the Cramér von Mises W due to Cramér and Mises [32] and the Kolmogorov Smirnov KS statistics due to Kolmogorov [33] with their p-values to compare the fitted models. These statistics are used to evaluate how a particular distribution with cdf, for a given data set, fits the corresponding empirical distribution. The distribution with better fit than the others will be the one having the smallest statistics and largest p-value. The descriptions of the data sets are as follows. The first data set is taken from engineering department consists of a sample of 3 failure times of airconditioned system of an aeroplane in hours and is presented by Linhart and Zucchini [34]. The data points are 23, 26, 87, 7, 2, 4, 62, 47, 225, 7, 246, 2, 42, 2, 5, 2, 2,, 3, 4, 7,, 4,, 6, 9,, 6, 52, 95. The second data is taken from the U.S. Geological Survey USGS gaging station 8235 Carnero creek near La Garita, Colorado which describes the annual maximum stream flow amount measured in cubic feet per second. The data is based on a time series data that consist of 58 periods from 92 to 98 data for 924, 925, 929 and 93 were missing. The data is available in U.S. Geological Survey USGS website http : //nwis.waterdata.usgs.gov. The data points are 44, 79, 5, 28, 8, 35, 32, 248, 59, 57, 278, 35, 327, 82, 86, 2, 3, 35, 435, 52, 75, 6, 66, 73, 239, 667, 44, 82, 7, 68, 69, 42, 6, 45, 333, 4, 2, 75, 299, 2, 93, 287, 64, 36, 438, 63, 46, 48, 37, 24, 25, 6, 4, 5, 32, 9, 28, 3 The third data describes the maximum rainfall in mm of the whole year of Jiwani town located along the Gulf of Oman in the Gwadar district of the Balochistan province in Pakistan from 98 to 2. The data points are 2.7, 72.9, 69.5, 96.5, 2.6, 265.5, 54, 28, 42.8, 4.2, 74.8, 32.5, 25, 28.5, 3.8, 25.7,

11 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 Table : Estimates of the parameters standard errors in parenthesis for Aeroplane data Distributions Estimates GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b Table 2: Goodness of fit statistics for Aeroplane data Distributions LogL AIC AICC HQIC CAIC GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b Distributions A W KS p GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b , 28, 6.9, 6, 9, 7.6, 47.3, 55, 29, 72, 92, 28, 3, 94 For the first data set, Table gives estimates of the parameters of the considered models with their corresponding standard errors. Table 2 presents their goodness-of-fit statistics. Concerning the GG model, the MLEs corresponding to the data are given by ˆθ =.554 and ˆλ =.24595, and the following information criterion are obtained: AIC = 36.25, AICC = , HQIC = 37. and CAIC = These values are the smallest in comparison to those obtained for the other models. On the other side, we have A =.5583, W = , KS =.473 with p = , which are also the best. The superiority of the GG model, in terms of goodness-of-fit statistics, in comparison to the others, is also observed for second data set estimates are given in Table 3 and goodness of fit statistics in Table 4 and the third data set estimates are given in Table 5 and goodness of fit statistics in Table 6. Acknowledgments Authors thank the reviewer for the suggestions for a better presentation. References [] Adamidis, K. and Loukas, S. 998: A lifetime distribution with decreasing failure rate, Statistics and Probability Letters, 39, [2] Adamidis, K., Dimitrakopoulou, T. and Loukas, S. 25: On a generalization of the exponential-geometric distribution, Statistics and Probability Letters, 73,

12 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 2 Table 3: Estimates of the parameters standard errors in parenthesis for CARNERO data Distributions Estimates GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b Table 4: Goodness of fit statistics for CARNERO data Distributions LogL AIC AICC HQIC CAIC GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b Distributions A W KS p GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b Table 5: Estimates of the parameters standard errors in parenthesis for Jiwani data Distributions Estimates GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b

13 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 3 Table 6: Goodness of fit statistics for Jiwani data Distributions LogL AIC AICC HQIC CAIC GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b Distributions A W KS p GGθ, λ Weibulλ, k Gumbelµ, σ EEα, λ GGuα, σ, µ Kappaα, θ, β WGλ, a, b [3] Alkarni, S. H. 22: A compound class of geometric and lifetime distributions, Open Statist. Probab. J. 5, -5. [4] Ahsanullah, M. 995: Record Statistics. Nova Science Publishers, Commack, New Jersey. [5] Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. 998: Records. Wiley, New York. [6] Barreto-Souza, W., Alice, L. M. and Cordeiro, G. M. 2: The Weibull-Geometric Distribution, Statistical Computation and Simulation, 8, [7] Bryson, C. and Siddiqui, M. M. 969: Some criteria for aging. Journal of the American Statistical Association, 64, [8] Castellares, F. and Lemonte, A. J. 26: On the Marshall-Olkin extended distributions. Communications in Statistics - Theory and Methods, 45, 5, [9] Chahkandi, M. and Ganjali, M. 29: On some lifetime distributions with decreasing failure rate, Computational Statistics and Data Analysis, 53, 2, [] Chandler, K. N. 952: The distribution and frequency of record values, Journal of the Royal Statistical Society: Series B, 4, [] Cordeiro, G. M. and Silva, R. B. 24: The complementary extended Weibull power series class of distributions, Ciência e Natura, 36, -3. [2] Harandi, S. S. and Alamatsaz, M. A. 27: A complementary generalized linear failure rate-geometric distribution, Communications in Statistics - Simulation and Computation, 46,, [3] Hollander, W. and Proschan, F. 975: Tests for mean residual life, Biometrika, 62, [4] Louzada, F., Roman, M. and Cancho, V. G. 2: The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart, Computational Statistics and Data Analysis 55, 8, [5] Louzada, F., Marchi, V. and Carpenter, J. 23: The complementary exponentiated exponential geometric lifetime distribution, Journal of Probability and Statistics, Article ID 5259, 2 pages. [6] Mahmoudi, E. and Jafari, A. A. 22: Generalized exponential-power series distributions. Computational Statistics and Data Analysis, 56, 2, [7] Mahmoudi, E. and Jafari, A. A. 27: The compound class of linear failure rate-power series distributions: Model, properties, and applications, Communications in Statistics - Simulation and Computation, 46, 2, [8] Morais, A. L. and Barreto-Souza, W. 2: A compound class of Weibull and power series distributions, Computational Statistics and Data Analysis, 553: [9] Muth, E. J. 977: Reliability models with positive memory derived from the mean residual life function, in Theory and Applications of Reliability, C. P. Tsokos and I. N. Shimi eds., Academic Press, [2] Rényi, A. 96: On measures of entropy and information, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley CA:University of California Press, I, [2] Silva, R. B., Bourguignon, M., Dias, C. R. B., and Cordeiro, G. M. 23: The compound class of extended Weibull power series distributions, Computational Statistics and Data Analysis, 58, [22] Tahir, M. H. and Cordeiro, G. M. 26: Compounding of distributions: a survey and new generalized classes. Journal of Statistical Distributions and Applications, 3,, -35. [23] Tahmasebi, S. and Jafari, A. A. 25: Exponentiated extended Weibull-power series class of distributions, Ciência e Natura, 37, [24] Ahuja, J. C. and Nash, S. W. 967: The generalized Gompertz-Verhulst family of distributions, Sankhya, Ser. A, 29, [25] Cooray, K 2: Generalized Gumbel distribution, Journal of Applied Statistics, 37, [26] Mielke P. W 973: Another Family of Distributions for Describing and Analyzing Precipitation Data, Journal of Applied Meteorology, 2 2, [27] H. Akaike 974: A new look at the statistical model identification, IEEE Transactions on Automatic Control, 9, [28] C. M. Hurvich, C. L. Tsai 989: Regression and time series model selection in small samples, Biometrika, 76,

14 Bakouch, Saboor, Chesneau, Saeed / ProbStat Forum, Volume 2, January 29, Pages 4 4 [29] E. J. Hannan, B. G. Quinn 979: The Determination of the order of an autoregression, Journal of the Royal Statistical Society, Series B, 4, [3] H. Bozdogan 987: Model selection and Akaikes information criterion, The general theory and its analytical extensions, Psychometrika, 52, [3] T. W. Anderson, D. A. Darling 952: Asymptotic theory of certain goodness of fit criteria based on stochastic processes, Annals of Mathematical Statistics, 23, [32] H. Cramer 928: On the composition of elementary errors II, Skand, Aktuarietidskr,, [33] A. Kolmogorov 993: Sulla determinazione empirica di una legge di distribuzione, G. Inst. Ital. Attuari, 4, [34] H. Linhart, W. Zucchini 986: Model Selection, Wiley, New York.